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	 \begin{table}[htbp] 
  \centering
  \caption{\bf Pricing Performance ($S\& P\, 500$):  Out-of-Sample -  Expected Returns  ($T=24$)}
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\begin{tabular}{lccccccccccc}
     \multicolumn{12}{c}{Out-of-Sample}   \\  \hline
   \# of factors:   &  $\tilde{r} $ & $1$  &  $ 2 $ & $3$ & $4$ & $5$ & $ 6 $ & $7$   & $8$  & $9$ &  $10$ \\   \hline
  constant-PCA  & - &   0.527 & 0.534 & 0.550 &  0.550 &  0.554 &  0.555 & 0.545 & 0.545 & 0.553 & 0.569 \\
  local-PCA (bal) & 0.600  &    0.517 & 0.576 & 0.621 & 0.671 & 0.687 & 0.701 & 0.717 &  0.734 &  0.739 & 0.743 \\
local-PCA (unbal) &  0.867  &  0.835 & 0.873 & 0.894 & 0.913 & 0.924 & 0.935 & 0.943 &  0.949 & 0.954 &  0.959 \\ \hline
   \multicolumn{12}{c}{In-Sample}   \\  \hline
   \# of factors:   &  $\tilde{r} $ & $1$  &  $ 2 $ & $3$ & $4$ & $5$ & $ 6 $ & $7$   & $8$  & $9$ &  $10$ \\   \hline
   \\ \hline
  constant-PCA &   - &  
 0.577 & 0.579 &  0.583 & 0.707 &  0.707 & 0.712 &  0.712 &  0.765 &  0.764 &  0.775 \\
 local-PCA (bal) &
  0.717 &  0.671 & 0.733 &  0.788 &  0.826 &  0.874 &  0.902 &  0.921 & 0.936 & 0.945 &  0.951 \\
 local-PCA (unbal) &  
 0.825 &  0.783 & 0.848 & 0.877 & 0.901 & 0.914 & 0.924 & 0.932 & 0.937 & 0.943 & 0.948 \\ \hline
    \end{tabular}
        \parbox{6.5in}{\footnotesize{The table shows the cross-sectional correlations between the average returns and the model-implied expected return estimated by 
        constant-PCA (first row), local-PCA balanced (second row), and local-PCA unbalanced (third row), when the number of risk factors ranges from one to $10$  (columns $3-12$), with the second column referring to the
      (time-varying) estimated number of factors $\tilde{r}$, described in Section~OA15.1.1. The average returns and model-implied expected returns are first evaluated over every rolling window of size $T=24$ and then averaged across all rolling windows. 
        The model-implied expected return is estimated out-of-sample.
 Data are monthly from Jan 1980 until Dec 2019.}}
\label{Table6OASP500}
\end{table}

	
	
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